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The properties *-regularity and uniqueness of \(C^*\)-norm in a general *-algebra. (English) Zbl 0532.46031

In this paper two important properties of a Banach *-algebra A are studied: the existence of a unique \(C^*\)-norm on A and the \({}^*\)- regularity of A. These properties are involved with the representation theory of A. For example when A is a reduced Banach \({}^*\)-algebra, then A has a unique \(C^*\)-norm if and only if every separating collection of \({}^*\)-representations of A lifts to a separating collection of \({}^*representations\) of the \(C^*\)-enveloping algebra of A. A number of basic results are derived concerning these important properties, and some applications are given in the case where \(A=L^ 1(G)\).

MSC:

46K10 Representations of topological algebras with involution
46H05 General theory of topological algebras
46L05 General theory of \(C^*\)-algebras
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